Probing Variations of the Rashba Spin-Orbit Coupling at the Nanometer Scale

Probing variations of the Rashba spin-orbit coupling at the nanometer scale Jan Raphael Bindel1, Mike Pezzotta1, Jascha Ulrich2, Marcus Liebmann1, Eugene Sherman3, and Markus Morgenstern1

1II. Institute of Physics B and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany

2 Institute for Quantum Information and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany

3Department of Physical Chemistry, the University of the Basque Country UPV-EHU and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

The Rashba effect as an electrically tunable spin-orbit interaction1 is the base for a multitude of possible applications2-4 such as spin filters3, spin transistors5,6, and quantum computing using Majorana states in nanowires7,8. Moreover, this interaction can determine the spin dephasing9 and antilocalization phenomena in two dimensions.10 However, the real space pattern of the Rashba parameter has never been probed, albeit it critically influences, e.g., the more robust spin transistors using the spin helix state6,11,12 and the otherwise forbidden electron backscattering in topologically protected channels.13,14 Here, we map this pattern down to nanometer length scales by measuring the spin splitting of the lowest Landau level using scanning tunnelling spectroscopy. We reveal strong fluctuations correlated with the local electrostatic potential for an InSb inversion layer with a large Rashba coefficient (~1 eVÅ). The novel type of Rashba field mapping enables a more comprehensive understanding of the critical fluctuations, which might be decisive towards robust semiconductor-based spintronic devices.

The Rashba effect1, which lifts spin degeneracy by breaking inversion symmetry at surfaces or interfaces, was firstly probed in transport using the beating pattern in Shubnikov-de Haas oscillations15 or the weak antilocalization effect.16 Later, Rashba–split bands and their spin polarization were visualized by photoelectron spectroscopy.17 The first successful attempts to use the Rashba effect for spin manipulation required low temperatures and found relatively small signals,18-21 probably due to D’yakonov-Perel’-type spin randomizing by disorder. Options to overcome this limit are more one-dimensional or ballistic devices.21,22 Another method is to balance the Rashba and the Dresselhaus couplings,6 leading to a persistent spin helix with a momentum 1 independent spin rotation axis.11 For such cases, where the D’yakonov-Perel’ mechanism is suppressed, other dephasing mechanisms, such as fluctuations of the Rashba parameter, limit device functionalities. Interestingly, the topological protection in spin channels of 2D topological insulators13,14 is also likely limited by fluctuations of the Rashba parameter in combination with electron-electron interaction13 or spin impurities.14

A natural method of investigating electronic disorder is scanning tunnelling spectroscopy

(STS).23-24 STS has already revealed fingerprints of the Rashba effect in two-dimensional electron systems (2DES) , such as a singularity at the band onset in the density of states (DOS)25,26, a beating pattern of the Landau levels in the DOS,27 or standing wave patterns from scattering between

Rashba bands and other spin degenerate bands26, thereby circumventing the absence of quasiparticle interference between the two components of the Rashba-split band.28 However, these methods did not probe the spatial fluctuations of the Rashba effect.

Here, we use an InSb 2DES, produced by Cs adsorbates on the (110) surface, to probe the

Rashba parameter  in real space. STS in magnetic field B reveals a nonlinear spin splitting of the

Landau levels (LL), which fits to the Rashba model at intermediate B = 3-7 T, while exchange enhancement29,30 dominates at higher B. Thus, the spin splitting at intermediate B can be used to trace  as a function of position R revealing that (R) spatially fluctuates between 0.4 eVÅ and

1.6 eVÅ. It exhibits a correlation length of 30 nm and a strong correlation with the electrostatic potential of the 2DES, as mapped as the spin-averaged LL energy.

The sample is sketched in Fig. 1a. By adsorbing Cs on p-type InSb(110), the valence and conduction bands are bent down towards the surface such that an inversion layer with one occupied subband is formed (Fig.1b).27 The Cs coverage (1.8% of a monolayer) is low enough to allow a

2 mapping of the 2DES by STS.27,29,31 A strong electric field |푬| ≈ 3 ⋅ 107 V/m within the 2DES results from acceptor-doping,27 which in combination with the large atomic numbers of In and Sb leads to a large . Figure 1c shows the spin-split LLs of the 2DES according to the Bychkov-

Rashba model.1 One recognizes crossing points of opposite spin levels (dashed ellipses) and a nonlinearity of the spin splitting at low B. Figure 1d shows this splitting for different , while keeping all other parameters identical. Different  naturally lead to different nonlinearities, offering an elegant method to locally determine . Although  is a strictly local parameter, the measured spin splitting is related to wave functions, such that the spatial resolution of the method is limited by about the cyclotron radius being smallest for the lowest level LL0. For smooth disorder potentials V(R) with respect to the magnetic length 푙B = √ℏ/푒퐵 (cyclotron radius of LL0), perturbation theory in 훁푉(푹)푙B/(ℏ휔푐) describes the energies n, adequately for different LLs n and spin labels  = +,:32

∗ 2 2 휆 푔푚 푉푛(푹)−푉푛−1(푹) 2√2훼(푹) 휀푛,휆(푹) = ℏ휔c (푛 − √(1 − + ) + 푛 ( ) ) + 2 2푚푒 ℏ휔c ℏ휔c푙B [1] 푉 (푹)+푉 (푹) 푛−1 푛 2

2 푉푛(푹) = ∫ 푉(푹 + 풓) ∙ 퐹푛(풓)푑 풓 [2]

th ∗ Fn(r) is the kernel of the n LL wave function (supplement S1) and 휔c = 푒퐵/푚 is the cyclotron frequency. We can therefore determine (R) from the measured splitting 휀0,− − 휀1,+, for known V(R), 푔 and 푚∗. 3

Figure 2a shows the DOS, i.e. the spatially averaged local density of states (LDOS) of the

2DES at B = 7 T. The characteristic beating pattern of the LLs found previously27 can be used to estimate the average Rashba parameter 훼̅ by comparison with the different fitting lines. The best agreement with the experimental beating pattern is found for 훼̅ = 0.7 eV ∙ Å in agreement with

27 ∗ previous results. For simplicity, we used a constant 푚 = 0.03 ⋅ 푚푒 and 푔 = −21 (averaged value, see below), neglecting the band nonparabolicity for the lowest LL. This causes the discrepancies at higher LLs. The width of the LL peaks is directly taken from the distribution of

V(R) (Fig. 3e). The observed strong dip at the Fermi level EF in the experiment is related to the well-known Coulomb gap.29,33

In order to extract the local Rashba parameter α(R), we recorded local LL fans. Figure 2b shows the measured LDOS of a single point at different energies and B. LLs and spin levels of two subbands are discernible as marked. The individual levels collectively undulate with B, which we ascribe to the undulation of all LLs with respect to EF in order to maintain the fixed carrier density and, to a lesser extent, to exchange enhancement.29 Reproducible instabilities in the spectroscopy are observed at distinct B (crosses, supplement Fig. S1c). Here, the conductance at EF partly drops to 3 pS, i.e. an insulating sample area close to integer filling factors prohibits current flow at these

29,33 B. We ascribe the slight suppression of LDOS around EF to a Coulomb gap.

Multiple crossings of levels are present, e.g. in the boxes marked I-III enlarged in Fig. 2c.

The dashed lines (guides to the eye) reveal that the marked levels cross away from B = 0 T, such that they cannot belong to simple Landau and spin energies, both being linear in B and crossing at

B = 0 T. A natural way to explain the crossings is the Rashba effect and, indeed, some of the crossings appear at rather similar B as in the calculations of Fig. 1c. Discrepancies, most obvious at lower B, are attributed to the local confinement within the potential minimum, where the data 4 are recorded. This complication hampers the use of the crossings for an accurate determination of

.

Instead, we use the nonlinearity of the LL0 spin splitting, Δ퐸푆푆 = 휀0,−(퐵) − 휀1,+(퐵).

Figure 2d shows the LDOS recorded at different positions at B = 14 T, exhibiting double peaks for

LL0 and more complicated structures for higher LLs. The complex peak structures appear away

31 from the extrema of V(R) due the nodal structure of the LL wave functions. The splitting ESS determined from fitting two Lorentzians to the two peaks of LL0 is indicated. It increases if the peaks shift to higher energies, i.e., at higher V(R). This is opposite to the expectation from the nonparabolicity of 푔(푉), which decreases with increasing V. Furthermore, a fluctuating peak width is observed, which discussion is beyond the scope of this manuscript.

Figure 2e shows Δ퐸푆푆(퐵) as deduced from Fig. 2b using Lorentzian fits (inset). Above B

= 8 T, ESS oscillates exhibiting maxima at odd filling factors as expected for exchange enhancement.29,30,33 Since the exchange interaction depends exponentially on the overlap of the wave functions which roughly scales with 푙퐵 ∝ √1/퐵 , it decays rapidly at lower fields, being

29 below 1 meV for B < 6 T. Accordingly, oscillations of ESS are barely discernible at B < 7 T.

Instead a largely linear ESS(B) is observed at 3-7 T decaying more rapidly at lower fields, similar to the curves of Fig. 1d. Extrapolating the linear part to B = 0 T (dashed line) reveals an offset of

0 31 Δ퐸푆푆~2.5 meV. Taylor expansion for large B of eq. 1 using 푉(푹) = 푐표푛푠푡., i.e. 푉푛(푹) = 푉(푹)

0 2 ∗ 2 ∗ reveals that the offset is given by Δ퐸푆푆(퐵 → 0) = 4훼 푚 /(ℏ (2 − 푔푚 /푚푒)) resulting in 훼 ≈

0.65 eVÅ close to 훼̅ = 0.7 eVÅ from Fig. 2a. Figure 2f shows the fitted ESS(B) curve for two different potential minima indicated by crosses in Fig. 3a-d. For the fit, we additionally consider the Vn terms of eq. 1 deduced from the potential V(R) of Fig. 3a (see below). The effective mass is

5 determined from the B-field dependent slope of the energy splitting between LL0 and LL1. The 푔-

∗ ∗ ∗ factor then results from the relation 푔(푉) ∙ 푚 (푉) = 푔0 ∙ 푚0, with 푔0 = −51 and 푚0 = 0.0135 ⋅

27 푚푒 at the conduction band minimum, leaving  as the only fit parameter. Neglecting the Vn terms would lead to slightly reduced  values (α(R) = 1.04 eVÅ (black cross) & 0.37 eVÅ (red cross)) as has been pointed out previously.32 More importantly, the obtained  values at those two positions differ by a factor of two indicating strong spatial  fluctuation. Both curves are recorded at potential minima in order to minimize influences by the spatial shape of V(R).

We can rule out the possibility that the spatial fluctuation of 푔 is responsible for the differences, since the curve probed at V(R) = -121 meV (black) should have a larger 푔-factor than that probed at V(R) = -101 meV (red) in contrast to the experimental observation. Exchange enhancement is also much weaker than the obtained differences in ESS, such that a spatially varying (R) is the most natural explanation.

Having established that (R) can be deduced from ESS, we produce (R) maps. A precise determination according to eq. 1 additionally requires Vn(R) maps, estimated by plotting the mean energy of the two LL0 levels (Fig. 3a). The resulting potential fluctuates by about  10 meV with a correlation length lVV = 50 nm. Convolving with the LL wave function kernel (eq. 2) leads to the

Vn(R) - maps in Fig. 3b, which are obviously smoother than Fig. 3a. Figure 3c shows the ESS(R) map at 6 T and Fig. 3d shows the resulting (R) map according to eq. 1. The ESS(R) and (R) maps exhibit a similar pattern, but differ in details.

Another contribution to (R) is the variation of 푚∗(푅) due to the nonparabolcity. We included this in Figure 3d according to 푚∗(푅, 푉) = 훽 + 훾 ∙ 푉(푅) where the effective masses directly deduced from Figs. S2a and b (Supplement) are used as anchor-points to determine  and 6

훾. This barely changes the spatially averaged 훼̅푠 (1 % reduction), but leads to an increase of the

FWHM of the distribution of (R) by 14 %, and a local change of (R) of even up to 30 %. Notably,

(R) fluctuates between 0.4 eVÅ and 1.6 eVÅ, i.e. by a factor of four. It exhibits a giant 훼̅푠 =

1.2 eVÅ, a rms fluctuation 훿훼 = 0.15 eVÅ (FWHM 30%) (Fig. 3e), and a correlation length of l

= 30 nm being larger than lB, but smaller than lvv (supplement S5). Therefore, (R) strongly fluctuates on small length scales.

It is known that (R) depends mostly on the local electric field E(R) perpendicular to the

2DES,1 which is not measurable by STM. However, we find a clear correlation of (R) with V(R), i.e., a larger V(R) implies a larger (R) (Fig. 3f). This is expected, since a larger V also implies a stronger local confinement, consequently, a larger E(R) as sketched in the insets of Fig. 3f. The remaining scatter of  at given V (휎̅훼̅ = 0.12 eVÅ), being much larger than the error bar of the  determination (휎̅훼 = 0.02 eVÅ), can be explained by the remaining scatter in the relation between

V and E (supplement S3 & S6).

Finally, we estimate the spin relaxation length lSpin from the data of Fig. 3d neglecting the

D’yakonov-Perel’ and the Elliott-Yafet mechanism, i.e., considering only the fluctuations of  for spin dephasing in 2D:34

2 ℏ2 1 푙Spin = ( ∗ ) ≈ 270 nm 2푚 훿훼 푙훼훼

The value, not limiting the spin relaxation in this particular system (supplement S8), can take over, e.g. if the D’yakonov-Perel’ mechanism is avoided as in ballistic21,22 or spin-helix6,11,12 transistors.

Consequently, a detailed understanding of the fluctuations of (R), as uniquely provided by our novel method, becomes crucial for these prospective devices.

Methods:

Preparation of clean InSb(110) surface InSb single crystals were glued with a conductive epoxy onto a Mo sample holder. A 1 mm deep notch was cut into the crystal to support cleaving along the (110) surface. A small screw was glued on top of the crystal. Inside an ultrahigh vacuum (UHV) chamber, at a base pressure of 10-10 mbar, the crystal was cleaved at the notch by pushing the screw towards the chamber wall. After in-situ transfer into a home-built STM within 1 hour and direct cooling to 9 K, 4 K, 1.5 K and 400 mK, respectively, atomically clean and flat terraces with a width of several µm were found.

2DES inversion layer 24 - The cleaved surface of the p-doped InSb with acceptor density NA = 1 ∙ 10 m ³ was transferred in UHV into a sample stage hold at T = 30 K, and Cs was evaporated from a Cs dispenser onto the surface. The Cs dispenser operated at 470°C contains cesium chromate. After three evaporation cycles of 180 s, the surface coverage is 1.8 % of a monolayer of Cs, defined as one Cs atom per InSb unit cell, as determined by counting the Cs atoms. During the whole procedure, the pressure did not exceed 1.6∙10-9 mbar. After the evaporation process, the sample was immediately transferred into the STM and cooled down to 1.5 K.

Peak fitting for determination of ε0,-, ε1,+, V(R), and spin splitting To discriminate the two spin levels of the lowest Landau level, we fit a double Lorentzian peak to the LDOS curves according to

푎 휎 푎 휎 LDOS(퐸) = 1 1 + 2 2 π 2 2 π 2 2 휎1 + (퐸 − 0,−) 휎2 + (퐸 − 1,+)

with amplitudes a1,2, peak widths 1,2 and peak energies 0,− and 1,+ for peak 1 and 2, respectively.

While for B > 3.5 T all six parameters are chosen to be free, for lower B, 1,2 were both fixed to 5.6 meV in order to deal with the less pronounced spin splitting. This is justified, since the distance of the two levels to EF barely changes leading to similar life times and, thus, similar peak widths.

To first order, the LLs probe the electrostatic potential with a resolution of about the cyclotron

32 radius. The electrostatic potential V(R) can, therefore, be estimated as the average of 0,− and 1,+ determined at the position R, while the spin splitting is the difference between 0,− and 1,+ .

Changes in the respective potential and in the spin splitting are not expected on a length scale shorter than lB(6 T) = 10.5 nm, hence we smoothed both maps with a Gaussian curve of width lB.

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Acknowledgements:

We appreciate helpful discussions with D. Hernangómez-Pérez, S. Florens, T. Champel, S. Becker, A. Georgi, C. Saunus, N.M. Freitag, M.M. Glazov, V.K. Dugaev and R. Winkler. Financial support by the German Science foundation via MO 858/11-2 and INST 222/776-1 is gratefully acknowledged. The work of E.S. was supported by the University of the Basque Country UPV/EHU under program UFI 11/55, Spanish Ministry of Economics and Competitiveness (grant FIS2012-36673-C03-01), and “Grupos Consolidados UPV/EHU” program of the Basque Country Government (grant IT472-10). J.U. acknowledges financial support via the Alexander von Humboldt-Stiftung.

Author contributions:

R. B. prepared the samples, conducted the experiments with the help of M. P. and M. L., evaluated the data with the help of M. L., and wrote the first version of the manuscript together with M. M.. E. S. supported the evaluation of the data and provided the analytic theory to determine Rashba disorder, spin dephasing and spin diffusion length. J. U. provided a theoretical analysis. M. M. and E. S. devised the overall idea of the experiment. All authors contributed to the interpretation of the data and revising the manuscript.

Figures: Fig.1

Figure 1: Rashba parameter from spin splitting of Landau levels. a, Sketch of the sample with the two-dimensional electron system (2DES) at the surface and conducting p-InSb bulk. An STM image of Cs/InSb(110) with atomic resolution showing Cs atoms (bright dots) on top of the lines of In atoms is displayed on the surface (20 nm  20 nm, V = 300 mV, I = 30 pA, T = 8 K). b, Band structure of the 2DES perpendicular to the surface, as resulting from a Poisson calculation.

Confined squared wave functions are sketched in yellow. Adsorbed Cs atoms act as donors. c, B- field dependence of the energy levels of one 2DES subband using 훼 = 0.7 eVÅ, 푔 = −21,

∗ 푚 = 0.03 ⋅ 푚푒 (average values from the potential area shown in Fig. 3a). Opposite spin 14 contributions are marked in blue and red. The Fermi level is shown as a dotted line, using 푛 =

1.5 ⋅ 1016 m−2. Crossing points of different spin levels are highlighted by dashed ellipses. Labels on the right are used in eq. 1. d, Splitting of the two lowest energy levels for different Rashba

∗ parameters 훼, 푔 = −21, 푚 = 0.03푚푒.

Fig.2

Figure 2: Local Rashba parameter deduced by STS. a, Circles: Spatially averaged differential conductance from 150150 nm² area (35  35 pixel) of Cs/InSb(110) (Vstab = 300 mV, Istab = 0.2 nA, Vmod = 3.5 mV, T = 400 mK, B = 7 T). Dotted lines mark E0 and E1, the onsets of the two different subbands. Full lines: Calculated density of states for different  as marked. Best

∗ correspondence of the beating antinode is found for 훼 ≈ 0.7 eVÅ (푚 = 0.03 푚푒, 푔 = −21) b, dI/dV measurement at a single position within a potential minimum of the 2DES recorded at changing B (Vstab = 50 meV, Istab = 0.1 nA, Vmod = 0.75 mV, T = 7.5 K). LLn mark the Landau levels of the lowest subband E0 and the first subband E1, n, mark levels of LLn according to eq. (1). The 16 local filling factor local, marked on top, results from counting all levels up to EF. Signs + mark areas of conductance at EF below 3 pS. These areas are artificially coloured blue, due to instability

(supplement, Fig. S1c). Black arrows mark the spin levels used in (e) and (f). Boxes with roman numbers mark the zoom regions shown in (c). c, Zooms into (b) with overlaid guides to the eye

(dashed lines), which follow two levels crossing at finite B. d, Differential conductance recorded at different positions (Vstab = 300 meV, Istab = 0.2 nA, Vmod = 0.7 mV, T = 400 mK, B = 14 T).

Landau levels LLn and spin splittings ESS, marked by horizontal double arrows, are indicated. The resulting spin splitting of LL0 is additionally marked on the left. Note that multiple peaks appear for higher LLs and that the sharpest LL0 levels are found at the lowest energies. e, Circles: energy difference of the two lowest energy levels of (b) (marked by black arrows in (b)). Dashed line

0 extrapolates the nearly linear slope between 3.4 T and 7.2 T towards 0 T as marked by E SS (B0).

Green and orange arrows belong to local = 5 and local = 7, respectively. Inset: dI/dV curves from low energy part of (b) at B as marked (dots) with fit line (violet) consisting of two Lorentzian peaks

(red, blue). The energy difference of the maxima of the two Lorentzians is shown as circles in the main image. f, Circles: LL0 splitting determined between 3 T and 6 T at two different positions

(red, black) as marked by crosses in Fig. 3a-d. Full lines: Fit according to the Rashba model with resulting local Rashba parameter α(R) marked. Dashed lines: 65% confidence interval of the fits with corresponding values of  α(R) marked.

Fig.3

Figure 3: Mapping of Rashba parameter and comparison with potential map. a, Map of the potential energy V(R) of the 2DES, which is (휀0,−(푹) + 휀1,+(푹)) /2 as resulting from fits of dI/dV data as described in methods and shown in the inset of Fig. 2e (Vstab = 50 meV, Istab = 0.1 nA, Vmod

= 1.5 mV, T = 400 mK, B = 6 T). Black and red cross show positions of curves in Fig. 2f drawn in the same colour. Black dotted square indicates the areas displayed in (b) and (d). The correlation length of the potential is lVV = 50 nm (supplement, Fig. S5c). b, V0(R) (left) and V1(R) (right) at B

= 6 T calculated from the respective area of the V(R) map according to eq. 2. c, LL0 splitting (ESS) map at B = 6 T. d, Spatially resolved Rashba parameter (R) determined by using eq. (1), the 2D potential from (a) and the spin splitting map from (c). e, histograms of the maps shown in (a), (b),

(c) & (d). Gaussian fits (red lines) are added with -values indicated. f, Black dots: (R) from (d) plotted as a function of V(R) from (a) always using the same position R. Red circles: averaged

Rashba parameter across a V(R) interval of 0.5 meV. Purple error bar indicates the typical error bars of V and  revealing that the scatter in  is significantly larger than the error bar. Insets: sketch of band bending at high local doping (top left) and low local doping (bottom right) with resulting local 2DES potential V and local E-field marked.

Supplementary information for “Probing variations of the Rashba spin-orbit coupling at the nanometer scale” Jan Raphael Bindel1, Mike Pezzotta1, Jascha Ulrich2, Marcus Liebmann1, Eugene Sherman3, and Markus Morgenstern1

1II. Institute of Physics B and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany

2 Institute for Quantum Information and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany

3Department of Physical Chemistry, the University of the Basque Country UPV-EHU and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

S1: Kernel functions for the determination of the effective Potential Vn

Deducing the Rashba parameter (R) from the measured spin splitting map requires additional knowledge on the electrostatic potential V(R), which has to be folded by the kernel of the Landau level wave functions (LLWFs) according to eq. 1 and 2 of the main text. To this end, the LLWFs can be rewritten as Ψn,R(r) with 푹 = (푋, 푌) being the center of a LLWF and 풓 = (푥, 푦) being the vector from this center to the point of interest1:

1 푧 − 푍 푛 |푧|2 + |푍|2 − 2푍푧∗ 훹 (풓) = ( ) exp (− ) 푛,푹 2 4푙2 √(2휋푙퐵푛!) √2푙퐵 퐵

Thereby, Z = X + iY and z = x + iy are complex coordinates of the center of the LLWF, often called the guiding center, and of the coordinate relative to this center, respectively. As described in the main text, lB is the magnetic length and n the LL index.

A straightforward calculation of the related Kernel of the LLWF results in:1

2 2 −(푙 /4)Δ푹 퐹푛(풓 − 푹) = 푒 퐵 |훹푛,푹(퐫)| 푛 2 (−1) 2(풓 − 푹) 2 2 −(풓−푹) /푙퐵 = 2 퐿푛 [ 2 ] 푒 휋푙퐵 푙퐵 with ΔR being the Laplacian with respect to R, Ln(z) the Laguerre polynomial of degree n, and 퐹−1(풓 − 푹) ≡ 0.

A transparent explanation for the inclusion of the resulting 푉푛(푹) into the spin splitting (eq. 1 of the main text) has been provided in reference 1. Due to the Rashba effect, the spin components become mixtures of different Landau levels, which are different for two adjacent spin levels, e.g., for ε1,+ and ε0,-. Since these LLWFs have different lateral extensions, the LLWFs of adjacent spin levels cover different areas of the potential map. Thus, they feel a different average electrostatic 20 potential, which shifts them in energy with respect to each other. Consequently, the spin splitting gets reduced within a potential minimum and enhanced within a potential maximum with respect to the result for a flat lateral potential. For the flat potential, one can show that 푉푛(푹) = 푉푚(푹) for all n and m, such that the influence of the potential on the spin splitting disappears.

S2: dI/dV B-field sweep

Figure S1: Mapping Landau level fans at different positions and temperatures. All dI/dV(V,B) spectra are measured within potential minima. The colour code shown at the lower right is valid for all three maps. Vstab = 50 meV, Istab = 0.1 nA, Vmod = 0.75 mV. a, Sweep from 6 T to 1 T, T = 400 mK. b, Sweep from 6 T to 1 T, T = 400 mK c, Sweep from 0 T to 14 T, T = 7.5 K. This map is the same as the one shown in Fig. 2b of the main text, but without removing the noise appearing at magnetic fields, where the sample resistance gets too large for the STM measurement (noisy stripes). 21

Figure S1a and S1b show additional dI/dV(V) data recorded while slowly ramping the magnetic field from 6 T to 1 T at T = 400 mK. In order to ensure that the spectra are recorded at the same position, the recording is interrupted after an increment of one Tesla. Then, an atomically resolved constant-current image is recorded and the tip is readjusted to the same position with respect to the adsorbates visible in the image. This procedure guarantees that a complete LL fan is recorded within 1 nm of the sample surface. The same procedure has been applied in Fig. 2b of the main text.

The resulting LL fan charts again reveal apparent crossings of adjacent levels at finite B as well as a suppression of intensity at EF reminiscent of the Coulomb gap expected for localized systems. While Coulomb gaps have been observed previously for n-type InSb samples, the crossings are 2 exclusive for the current p-type sample with large Rashba coefficient. The spin splitting of LL0 of these maps has been used for Figure 2f of the main text with a (b) related to the black (red) curve. Figure S1c again shows the spectra of Figure 2b, but now without removing the regions of unstable tunnelling conditions.

S3: Relation between potential V(R) and electric field E(R) In Fig. 3f of the main text, it is shown that (R) on average increases monotonously with V(R). However, the scattering of (R) at the same V(R) is much larger than the error bar of (R) (see supplement S6). This is related to the fact that (R) is known to be proportional to the electric field |E(R)| (Fig. 1b), which is only indirectly represented by the measurable V(R) (inset of Fig. 3f, main text). In order to estimate the correlation between electric field and potential, a simple numerical model is applied. Therefore, we distribute positively charged Cs atoms on the surface, which are compensated by randomly distributed negatively charged acceptors within the bulk. Figure S2 shows such a random distribution of negative charges (red dots) with a density of 1024 m-3, which corresponds to the experimental acceptor density. The surface doping is realized by equidistant positive charges (4 nm distance, green dots) mimicking the fact that much more Cs atoms are deposited than electrons are transferred to the InSb, such that the remaining positive charges can arrange rather regularly within the Cs layer.2 We simply add up the Coulomb potentials of the positive and negative charges within a plane which is offset by 5 nm from the plane where the acceptors and surface donors are placed. This approximates the average situation of potential fluctuations within a 3D material by removing the measurement plane by about half the inter-acceptor distance from the acceptor plane. As dielectric 3 constants, we use 휖푟 = 휖푟,InSb = 16.8 for the negative charges, and 휖푟 = 0.5 ⋅ 휖푟,InSb for the positive charges. The latter choice reflects that the Cs atoms are surrounded by vacuum in the upper half-space. Screening itinerant carriers is ignored for the sake of simplicity. The resulting electrostatic potential, respectively, the corresponding electric field in z-direction of the inner part of the simulation grid is shown in Fig. S2a. Both, electric field and potential, are finally convolved

22 with the electron probability density distribution of the lowest subband of a triangular potential4 adapted to the result of the Poisson calculation, with its maximum located at 푧0 = 5 nm. We checked that the resulting relative fluctuation of the electric field does barely depend on the spatial details of the dielectric constant. For each simulation run, we use only five equidistant x,y positions within the inner 50 nm of the measurement plane for the statistical evaluation.

Figure S2b shows the scatter plot of the resulting effective electric fields |푬푧,eff| and effective potentials 푉푧,eff for all evaluated lines. Besides the expected linear dependency of the average electric field on the effective potential (red circles), a significant scattering of the data points is present. The standard deviation of the |푬푧,eff| distribution at given 푉푧,eff is ~10%, which matches nicely to the standard deviation of the Rashba parameter distribution at given 푉푧,eff shown in Figure 3f of the main text. Therefore, we tentatively assign the observed large scatter of (푉푧,eff) to the unavoidable scatter of |푬푧,eff|(푉푧,eff) within a random potential. The absolute value of the effective ̅̅̅̅̅̅̅̅ 7 electric field in our strongly simplified 2D simulation ( |푬푧,eff| = 1.3 ⋅ 10 V/m) is smaller than the electric field of |푬| = 3.1 ⋅ 107 V/m resulting from the Poisson-Schrödinger equation, but that 5 does not affect the general conclusion of the unavoidable scatter of |푬푧,eff|(푉푧,eff).

Figure S2: Interplay between 2DES potential and electric field. a, Electrostatic potential V (left) and electric field in z-direction |E| (right) resulting from randomly distributed, negatively charged 24 -3 2/3 acceptors (red dots) with density (ρZn = 10 m ) , negatively charged surface donors with density -1 ρCs = 0.25 nm (green dots). The area of simulation is larger than the displayed one (x = 150 nm, z = 100 nm). b, Correlation between effective potential and effective electric field (after folding with the wave function of the first subband of the 2DES) for 2500 simulated different x-positions (black dots) resulting from 500 simulations as displayed in (a). Red dots mark the average electric field

Ez,eff for a given 푉푧,eff within an interval of  0.25 meV.

S4: Correlation of the Rashba parameter with lateral gradient and curvature of the electrostatic potential

Figure S3: Correlation of lateral gradient and curvature of the electrostatic potential with the Rashba parameter. Correlation plots of the experimental data shown in Fig. 3 of the main text. Black dots indicate the data points and red circles the average values. a, Correlation between the lateral electric field and the local Rashba parameter. b, Correlation between curvature of the 2DES potential and the local Rashba parameter. c, Correlation between curvature of the potential and the local Rashba parameter after subtracting the average Rashba parameter found for the corresponding local potential.

In Fig. 3f of the main text, we have shown that (R) increases monotonously with the electrostatic potential, which is explained by the relation between potential and vertical electric field (Fig. S2b). Due to the presence of the potential disorder, there exists also a lateral electric field Exy, which is the lateral gradient of the local potential V(R). Figure S3a shows the absence of correlation between (R) and |Exy|. The lateral electric field has negligible influence on the Rashba parameter, which is reasonable, since |Exy| is about an order of magnitude smaller than the vertical electric fields. In contrast, the curvature of the potential appears to anticorrelate with the Rashba parameter (Fig. S3b). But the curvature itself is not uncorrelated from the potential, i.e. potential minima (maxima) show large positive (negative) curvature. In order to disentangle this indirect effect between curvature and (R) from a direct influence, we subtract the average value of  found for the particular 푉푧,eff(푹) (red dots in Fig. 3f of the main text) from the measured (R). The result is shown in Fig. S3c. The correlation obviously disappears. Therefore, as expected, neither the small lateral electric field nor the lateral potential curvature influence the Rashba effect significantly leaving the spatially fluctuating E-field perpendicular to the surface as the central influence on the variable Rashba parameter α(R).

S5: Correlation lengths of V(R) and α(R)

Figure S4: Auto- and cross-correlations of electrostatic potential map and map of the Rashba parameter. Correlation maps of the experimental data shown in Fig. 3 of the main text. a, b, Autocorrelation of the potential map (a) and the Rashba parameter map (b). The correlation length in ⟨훼|훼⟩ is smaller than in ⟨푉|푉⟩. c, Radial average of panels (a) (black circles), (b) (red crosses), and (d) (blue squares). The resulting correlation lengths are 퐿푉푉 = 49 nm, 퐿훼훼 = 31 nm, and are 퐿푉 = 39 nm. Note that the cross correlation ⟨푉|훼⟩ at 0 nm amounts to ~ 0.6. d, e, f, Cross- correlations of potential and Rashba parameter map, lateral electric field |∇푉| and Rashba parameter map, as well as potential curvature Δ푉 and Rashba parameter map as marked. Scale bars of all panels are 50 nm. Colour code is identical for (a), (b), (d), (e), and (f).

Figure S4 presents different autocorrelation and cross-correlation maps between (R), 푉푧,eff(푹), and the gradient and curvature of 푉푧,eff(푹). By angularly averaging the correlation maps (Fig. S4c), the correlation length 퐿퐴퐵 can be determined according to: 푆 2 퐿퐴퐵 = 2휋 ∫ 푓퐴퐵(푥)푥 푑푥. 0 The upper integration limit S is set as the distance at which the radial averaged function crosses zero for the first time (SVV = 50 nm, Sαα = 40 nm, SVα = 40 nm). The dimensionless function 푓퐴퐵(푥), with 푓퐴퐵(0) = 1, is given by a Gaussian fit to the radial averaged curve up to this point, with the width σi as the only free parameter. This results in σVV = (20.1 ± 0.7) nm, σαα = (12.2 ± 0.2) nm, and σVα = (16.0 ± 0.5) nm. For the area shown in Fig. 3 of the main text we obtain 퐿푉푉 = 49 nm, 퐿훼훼 = 25

31 nm, and 퐿푉훼 = 39 nm, all being larger than the magnetic length lB = 10.5 nm. The fact that α(R) is fluctuating on a shorter length scale than the potential disorder is not completely understood, but might be related to different fluctuation lengths of 푉푧,eff(푹) and |Ez,eff(R)|. The cross- correlation map ⟨푉(푹)|훼(푹)⟩ shows a relatively strong interrelation up to 60%, which becomes, moreover, manifested by the similar features of the real space maps (Fig. 3 main text) and autocorrelation maps of V(R) and α(R).

Cross correlations of (R), with the gradient and the curvature of 푉푧,eff(푹) are shown in Fig S4e and f. They exhibit rather weak features, which, however, are not analyzed in detail.

Generally, the simultaneous mapping of α(R) and 푉푧,eff(푹) of a 2DES by our novel method opens ample possibilities for a detailed analysis, e.g. via correlation maps.

S6: Errors on the determination of spin splitting and α(R)

Figure S5: : Histograms of errors. a, Error histogram of the spin splitting determination by the fits as shown in Fig. 2e of the main text. The data leading to the spin splitting map in Fig. 3c of the main text are used. 65% of the errors are smaller than σdE = 0.5 meV (red line). b, error histogram of the resulting local Rashba parameters. 65% of the errors are smaller than σα = 0.02 eVÅ (red line). The error on the determination of m* (5 %) is not included, since this error results in an uncertainty of 훼̅, but not in uncertainties of the relative –values at different positions, since the

26 same relation between V and 푚∗ is used for the whole image. c, error on α with included error on m*. The 65% percentile is σα = 0.04 eVÅ (red line).

The error on the determination of the spin splitting (σΔESS) and the local potential value (σV) is both 0.5 meV (65% percentile). It results directly from the peak fitting of the double Lorentzians (Figure S5a). To determine α(R), firstly, the potential map and the spin splitting map are Gaussian filtered (3x3 pixel, width 10 nm). This reduces the error of α(R) to σα = 0.02 eVÅ (65% percentile). The corresponding distribution is shown in Fig. S5b. Due to the nonparabolicity of the dispersion relation, we have to make assumptions for m*, which influence the accuracy of the determination of the absolute value of α(R). We assume that m* is linear in E, which is a simplification, since the evaluated energies relative to the conduction band minimum are larger than the band gap.6 The linearity is fixed by the two m*, which are determined at two fixed positions with known potential by looking at the B-field dependence of the LL splitting * ∗ Δ퐸퐿퐿 at B = 3-6 T (m = 0.030me & 0.026me). Here, we use Δ퐸퐿퐿 = ℏ푒퐵/푚 neglecting the * ∗ contribution of the Rashba effect, which results in an m error of 5 %. (α = 1 eVÅ, 푚 ≈ 0.028푚푒). By this, we estimate the error on the effective mass to σm* = 0.002 me. This error is an order of magnitude larger than the fitting accuracy of the LL splitting. However, with the inaccuracy of the linear model in mind, we refrain from a recursive more accurate determination of m*. As described ∗ ∗ in the main text, the local Landé-factor 푔 is determined according to 푔(푉) ∙ 푚 (푉) = 푔0 ∙ 푚0, with ∗ 7,8 푔0 = −51 and 푚0 = 0.0135푚푒 increasing the influence of this error additionally. Figure S5c shows the distribution of errors on α(R), taking the uncertainty of 푚∗(푉) and 푔(푉) into account. Thus, the absolute accuracy of α(R) is estimated to σα  0.04 eVÅ (4 %), while the relative accuracy at different spatial positions is ~2 %.

S7: Expected fluctuations of the Rashba coupling due to random acceptors and positive Cs charges Here we present analytical estimates of the random Rashba coupling and corresponding correlation lengths based on the approach developed in detail in Ref. [9]. In this model, fluctuations in the spin-orbit coupling appear due to random electric fields of the negatively charged acceptors in the depletion layer and the positively charged Cs ions at the surface. As the distribution of the random charge density, we take a similar one to that considered in section S3. We take the electron probability density distribution (see Fig. 1(b) of the main text) |Ψ(푧)|2 as:4 |Ψ(푧)|2 = (푏3/2) ⋅ 푧2 exp(−푏푧), -1 with the maximum position at z0 = 5 nm, corresponding to b = 0.4 nm . The local value of the spin- orbit coupling is produced by averaging the z-component of the electric field over the given probability density |Ψ(푧)|2. As a result of having two sources of the electric field fluctuations, we obtain two correlation functions of the Rashba parameters: 2 2 〈훿훼2D(ퟎ)훿훼2D(푹)〉 = 〈(훿훼2D) 〉퐹2D(푅), 〈훿훼3D(ퟎ)훿훼3D(푹)〉 = 〈(훿훼3D) 〉퐹3D(푅) 27

Here, F2D and F3D are the corresponding range functions. They can be calculated straightforwardly using lengthy integral expressions for correlation functions. The subscript 2D is used for the contribution of the surface Cs with the mean density 푛̅2D and 3D is used for that of the bulk acceptors with the mean density 푛̅3D, and R is the 2D coordinate. We assume here an uncorrelated white-noise distribution of dopants, both, on the surface and in the bulk, described by:

〈푛2D(ퟎ)푛2D(푹)〉 = 푛̅2D훿(푹), 〈푛3D(ퟎ)푛3D(풓)〉 = 푛̅3D훿(풓), [1] where 풓 is the 3D coordinate. These densities of charged particles produce at a point 풓 a random electric field with unscreened z-component 2 in the form

1 풓 − 푹 풓 − 풓′ 퐸 (풓) = 푒 [2 ∫ 푛 (푹) 푑2푅 − ∫ 푛 (풓′) 푑3푟′] , 푧 휖 2D |풓 − 푹|ퟑ 3D |풓 − 풓′|ퟑ r,InSb 푧 where 푑3푟′ integration is taken over the depletion layer. Analog to supplement S3 we assume that the dielectric constant is 휖푟 = 0.5 ⋅ 휖푟,InSb for positive and 휖푟 = 휖푟,InSb for negative charges. Details of the coordinate dependence of the dielectric constant do not have a considerable effect on our results. 2 By averaging the products 〈퐸푧(풓1)퐸푧(풓2)〉 over the disorder by using the distribution |Ψ(푧)| and the above presented white-noise correlators of the concentrations, one obtains9, after a lengthy but straightforward calculation, the correlation functions of the random contribution to the Rashba parameters as presented above. The variations of the spin-orbit coupling have the form: 2 1/2 2 1/2 〈(훿훼2D) 〉 = (2/휖푟 ⋅ 푒 휉) × 푏(휋/10 ⋅ 푛̅2D) , [2] 2 1/2 2 1/2 〈(훿훼3D) 〉 = (1/휖푟 ⋅ 푒 휉) × (휋/2 ⋅ 푛̅3D푏) , [3] where the material parameter 휉 describes the proportionality between the electric field and the Rashba parameter.6 The corresponding correlation lengths are defined similarly to the supplement S5, using the functions 퐹2D(푅) and 퐹3D(푅) : ∞ 2 퐿2D(3D) = 2휋 ∫ 퐹2D(3D)(푅)푅푑푅. 0 This results in: 40휋 휋 퐷 퐿2 = , 퐿2 = ⋅ 푙, 2D 푏2 3D 2 푏 where Dl is the depletion layer depth. Note that since the -functions of equation [1] do not have a characteristic nonzero spatial scale, the correlation length of the electric fields produced by the 2 surface charges, 퐿2D, depends solely on the width of |Ψ(푧)| , that is, 1/b. However, the bulk dopants are characterized, in addition, by the Dl spatial scale. As a result, 퐿3D includes two length parameters and increases relatively slowly with Dl as the far-distant ions produce relatively weak 2 1/2 fluctuations of the electric fields, resulting in the independence of 〈(훿훼3D) 〉 of the depletion layer depth in the 퐷푙푏 ≫ 1 limit applied to the derivation of equation [2] and corresponding to our 28 system parameters. It is interesting to mention that the correlation lengths depend solely on the system geometry and not on the material and sample parameters such as 휖푟, 휉, and charge densities. 2 16 −2 Substituting the characteristic numbers: 휉 = 526 Å (as in Ref. [6]), 푛̅2D = 6 × 10 m , 푛̅3D = 24 −3 10 m , 휖푟 = 16.8 for the dielectric constant of InSb, and Dl = 30 nm for the depletion layer 2 1/2 2 1/2 width we obtain 〈(훿훼2D) 〉 ≈ 〈(훿훼3D) 〉 ≈ 0.25 eVÅ for the variations and L2D ≈ 28 nm, L3D ≈ 11 nm for the correlation lengths, respectively. These numbers, being approximate, show a reasonable agreement with the experiment ( 훿훼 = 0.15 eVÅ, 퐿훼훼 = 30 nm) and allow us to attribute the experimental results to the fluctuations in the electric fields produced by randomness in the distribution of dopants. Notice that the variation in the Rashba parameter due to the acceptors is smeared out in the experiment since L3D is close to the value of the magnetic length.

S8: Comparison of spin dephasing by fluctuating Rashba parameter and D’yakonov-Perel’ mechanism Here we present the estimates of characteristic values of the spin diffusion length due to the regular and the random spin-orbit couplings of our system. Taking the electron concentration 푛 = 16 −2 8 −1 10 m , we obtain the Fermi wave vector 푘F = √2휋푛 = 2.5 × 10 m and the corresponding 6 ∗ velocity 푣F = 10 m/s for 푚 = 0.03 ⋅ 푚푒. For obtained in the main text 훼̅ = 1 eVA, the regular 14 −1 10 3 2 spin precession rate is Ω푆푂 = 2훼̅푘F/ℏ ≈ 10 s . A typical mobility 휇 = 10 cm /Vs yields the −14 electron mean free path l = 15 nm and momentum relaxation time휏 = 푙/푣F = 1.5 × 10 s. The product Ω푆푂휏 ∼ 1 implies that the spin relaxation, being caused by the electron momentum randomization, is, however, not described by the conventional D’yakonov-Perel’ diffusion-like formula, and the spin relaxation time is of the order of the momentum relaxation time 휏.11 As a [reg] result, the corresponding spin diffusion length 푙푠 = √퐷휏 is close to the free electron path l, with 2 D as the diffusion constant given by 퐷 = 푣F 휏/2. 7 The spin-orbit coupling correlation length 푙 = 30 nm yields for 훿훼 = 0.15훼̅ the random 2 12 −1 contribution to the spin relaxation rate Γrnd ≈ (2훿훼푘F/ℏ) 푙훼훼/푣F ∼ 2 × 10 s and spin −1 −13 [rnd] relaxation time: Γrnd ∼ 5 × 10 s ∼ 30휏 . The corresponding spin diffusion length 푙푠 = −1 [reg] [reg] √퐷Γrnd is about 5푙푠 , thus, larger than 푙푠 . Note that the spin relaxation length lSpin as given in the main text describes the traveled path of the electron prior to relaxation and, thus, is larger than [rnd] 푙푠 .

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